A334412 Number of ordered pairs of permutations of [n] avoiding synchronous double descent pairs.

1, 1, 4, 35, 545, 13250, 463899, 22106253, 1375915620, 108386009099, 10540705282001, 1240370638524842, 173704235075714947, 28549174106487593365, 5441843626292088857818, 1190762128123996264128849, 296456799935194225886732961, 83321234634397591315509479058

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100

Wikipedia, Permutation

Formula

a(n) <= A001044(n) with equality only for n < 3.

Example

a(3) = (3!)^2 - 1 = 35: only (321,321) does not avoid synchronous double descent pairs among the ordered pairs of permutations of [3].

Maple

b:= proc(n, u, v, t) option remember; `if`(n=0, 1, add(add(
      b(n-1, sort([u-j, v-i])[], 1), i=1..v)+add(
      b(n-1, sort([u-j, v+i-1])[], 1), i=1..n-v), j=1..u)+add(add(
      b(n-1, sort([u+j-1, v-i])[], 1), i=1..v)+add(`if`(t=0, 0,
      b(n-1, sort([u+j-1, v+i-1])[], 0)), i=1..n-v), j=1..n-u))
    end:
a:= n-> b(n$3, 1):
seq(a(n), n=0..21);

Mathematica

nn = 20; a=Apply[Plus, Table[Normal[Series[y x^3/(1 - y x - y x^2), {x, 0, nn}]][[n]]/(n +2)!^2, {n, 1, nn - 2}]] /. y -> -1; Map[Select[#, # > 0 &] &,
Range[0, nn]!^2 CoefficientList[Series[1/(1 - x - a), {x, 0, nn}], {x, y}]] // Flatten (* Geoffrey Critzer, Apr 27 2020 *)

Crossrefs

Column k=0 of A334257.

Cf. A000275 (the same for single descent pairs), A001044.

Sequence in context: A165933 A005973 A007134 * A238390 A251591 A346802

Adjacent sequences: A334409 A334410 A334411 * A334413 A334414 A334415

Keyword

nonn

Author

Alois P. Heinz, Apr 27 2020

Status

approved